Angular momentum is a measure of the quantity of rotation of an object and depends on the distribution of the mass of the object and the rate at which it is rotating. It can be thought of as the rotational equivalent of linear momentum and plays a similar role in rotational dynamics as linear momentum does in linear motion.
The angular momentum 𝐿 of a rotating object can be mathematically expressed as:
𝐿=𝐼𝜔
Where:
𝐼 s the moment of inertia of the object, which measures how the mass is distributed relative to the axis of rotation. The greater the mass is distributed away from the axis, the larger the moment of inertia.
𝜔 (Greek letter omega) represents the angular velocity, or how fast the object is spinning. It is typically measured in radians per second
0. Rank the following from largest to smallest angular momentum.
Ans: 5>1=4>2=3
A solid disk with a radius of 0.5 meters is spinning at 6 radians per second. The moment of inertia for a solid disk can be found in the table below, and the mass of the disk is 4 kg, calculate the angular momentum of the disk. ANS: 3 kg ●m2/s
2. A spinning bicycle wheel has an angular momentum of 24 kg·m²/s and a radius of 0.3 meters. If the wheel rotates at a rate of 8 radians per second and the moment of inertia is given in the table below, find the mass of the wheel. ANS: M= 33.3 kg
3. A cylindrical object has a mass of 10 kg, rotates at 2 radians per second, and possesses angular momentum of 5 kg·m²/s. Given the moment of inertia of a cylinder about its central axis from the table below, determine the radius of the cylinder. ANS: R=0.707m
4. A flywheel begins to spin up from rest and 2 minutes later, it has an angular momentum of 720 kg·m²/s. Assuming a constant torque was applied, determine the additional time for the flywheel to reach an angular momentum of 1080 kg·m²/s at the same rate of increase. ANS: additional 60s
Table of Rotational Inertia Values
The Law of Conservation of Angular Momentum is a fundamental principle in physics that states the total angular momentum of a closed system remains constant if no external torques are applied. This concept is crucial for understanding rotational dynamics in various physical systems, from spinning toys to orbiting planets.
In practice, the law of conservation of angular momentum means that if the total external torque acting on a system is zero, the total angular momentum of the system cannot change. This conservation can be observed in everyday phenomena such as a figure skater spinning faster when they pull their arms in close to their body. By reducing their moment of inertia, their angular velocity increases to keep the angular momentum constant in the absence of external torques.
The Law of Conservation of Linear Momentum is a foundational principle in physics that asserts the total linear momentum of a closed system remains constant if no external forces are acting on it. This law is central to analyzing and understanding the behavior of objects in motion in various physical contexts, from collisions to rocket propulsion.
In practical terms, the conservation of linear momentum implies that if two or more objects interact in a system where no external forces interfere (like friction or air resistance), the total momentum before the interaction will equal the total momentum after the interaction. This conservation is evident in phenomena such as collisions between objects. For instance, in a perfectly elastic collision where two billiard balls collide, the total momentum of the balls before and after the collision remains unchanged, though their individual momenta might change.
In an isolated system, the moment of inertia of a rotating object is halved. What happens to the angular velocity of the object? ANS: ω doubles
2. A satellite is rotating once per minute. It has an moment of inertia of 10,000kg⋅m2 . Erin, an astronaut, extends the satellite's solar panels, increasing its moment of inertia to 30,000 kg⋅m2 . How quickly is the satellite now rotating? ANS: ω = 0.035 rad/s or T=180s/ rev
4. A flywheel rotates without friction at an angular velocity ω0 =600 rev/min on a frictionless, vertical shaft of negligible rotational inertia. A second flywheel, which is at rest and has a moment of inertia three times that of the rotating flywheel, is dropped onto it. Because friction exists between the surfaces, the flywheels very quickly reach the same rotational velocity, after which they spin together.
(a) Use the law of conservation of angular momentum to determine the angular velocity 𝜔 of the combination. Ans: 15.7 rad/s
(b) What fraction of the initial kinetic energy is lost in the coupling of the flywheels? Ans: 3/4 KE is LOST
5. An 80.0-kg gymnast dismounts from a high bar. He starts the dismount at full extension, then tucks to complete a number of revolutions before landing. His moment of inertia when fully extended can be approximated as a rod of length 1.8 m and when in the tuck a rod of half that length. If his rotation rate at full extension is 1.0 rev/s and he enters the tuck when his center of mass is at 3.0 m height moving horizontally to the floor, how many revolutions can he execute if he comes out of the tuck at 1.8 m height? ANS: 2 revolutions at 4.0 rev/s
6. An ice skater spins with a specific angular velocity. She brings her arms and legs closer to her body, reducing her rotational inertia to half its original value. What happens to her angular velocity? What happens to her rotational kinetic energy?
As the skater pulls her arms and legs in, she reduces her moment of inertia. Since there is no external net torque, her spin angular momentum remains constant; therefore her angular velocity must double. Rotational kinetic energy, on the other hand, is governed by K=½Iω2. Moment of inertia is cut in half, but angular velocity is doubled; therefore rotational kinetic energy is doubled. The skater does work in pulling her arms and legs in while spinning.
AP Student Edition Problems